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The computational basis is orthonormal

Statement

For every $d \geq 1$ , the ordered computational basis $(e_{0}, \dots, e_{d - 1})$ of the register $Q_{d}$ from Finite registers and their ordered computational bases is an orthonormal basis for its declared inner product. Separately, the one-dimensional zero-qubit register has the basis ${1}$ by definition.

Facts & Assumptions

Given: A positive integer $d$ , the coordinate space $C^{d}$ , and its standard pairing.

[F1]

The standard pairing on $C^{d}$ is $⟨ z, w ⟩_{r o o t} = \sum_{r < d} z_{r} \overline{w_{r}}$ and is linear in the first argument (The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products).

[F2]

The standard coordinate vectors form an ordered basis of $C^{d}$ , including the empty basis for $C^{0}$ (The standard list $e : n \to F^{n}$ with $e_{i} (i) = 1_{F}$ and $e_{i} (j) = 0_{F}$ for $j \neq i$ is an ordered basis of $F^{n}$ ; hence $\dim_{F} F^{n} = n$ , and $F^{0}$ is the zero space with basis $\emptyset$ and dimension $0$ ).

[F3]

The register $Q_{d}$ uses this ordered standard basis and this pairing. Separately, the zero-qubit register is the empty tensor product $C$ with basis ${1}$ , distinct from $C^{0}$ (Finite registers and their ordered computational bases).

Proof

1.1givenF1F3algebra

For $j, k < d$ , the coordinate formula in [F1] gives $⟨ e_{j}, e_{k} ⟩_{r o o t} = \sum_{r < d} δ_{r j} δ_{r k}$ , which equals $1$ when $j = k$ and $0$ when $j \neq k$ . Thus the listed vectors are orthonormal.

2.1step 1.1F2F3

By [F2] the same vectors form a basis, so they are an orthonormal basis of $Q_{d}$ .

3.1F3∎

By [F3], the zero-qubit register is the empty tensor product $C$ with basis ${1}$ , distinct from the zero-dimensional coordinate space $C^{0}$ .

Depends on

Used by

Dense-coding procedure Algorithm
Gray-Path Synthesis of a Two-Level Gate Algorithm
One-query Bernstein–Vazirani algorithm Algorithm
One-query Deutsch algorithm Algorithm
One-query Deutsch–Jozsa algorithm Algorithm
Qubit teleportation procedure Algorithm
Reversible embedding of a Boolean circuit Algorithm
A preserved unknown state cannot program its own reflection Counterexample
Generator convention matters Counterexample
One uniform phase query gives small trace-distance advantage Counterexample
Binary phase-query convention Definition
Boolean oracle query model for search lower bounds Definition
Clean-ancilla zero-branch postselection Definition
CNOT and Control-Target Order Definition
Coherent XOR function oracle Definition
Controlled Phase Rotation Definition
Controlled-Z and SWAP Definition
CSS code from nested binary codes Definition
Normalized output law for finite quantum query branches Definition
Oblivious ancilla reflection Definition
Qubit spaces and states Definition
Query-address projectors for bit-oracle algorithms Definition
Query-register projectors and mismatch indicators Definition
SDP maximally entangled reference-state access Definition
Toffoli and Fredkin gates Definition
Two-Level Unitary Definition
A reversible AND oracle with clean work Example
A three-stage precision budget Example
Bloch coordinates of |0> and H|0> Example
Complete-graph search matrix and gap Example
Controlled Global Phase Changes an X-Outcome Example
Finite hybrid budget for a four-address search space Example
Promise, query and cost-model examples Example
Register endianness and a two-qubit partial trace Example
Small hypercube transfer Example
Two-term LCU postselection is input dependent Example
Acceptance is invariant under an unconditional global phase Lemma
Clean and dirty ancilla identities extend to references and mixed states Lemma
Clean reversible controlled multiplication by a cyclic constant Lemma
Coherent normal form for finite quantum query algorithms Lemma

…and 70 more results.

Cited to discharge well-definedness by Finite registers and their ordered computational bases.

Dependency tree · two levels

29 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Watrous, The Theory of Quantum Information (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes (standard reference, not scraped)